 # Resources for Teaching Discrete Mathematics: Classroom Projects, History Modules, and Articles

Edited by Brian Hopkins
Series: MAA Notes
Volume: 74
Edition: 1
Pages: 338
https://www.jstor.org/stable/10.4169/j.ctt5hh8fd

1. Front Matter
(pp. i-vi)
2. Introduction
(pp. vii-x)

For some twenty years now, the MAA Notes Series has published secondary materials for undergraduate mathematics courses, such as projects that can be used in teaching calculus. These publications reflect the interests of instructors, providing a means of sharing innovative ideas for teaching calculus, linear algebra, differential equations, statistics, geometry, and abstract algebra. With this book, discrete mathematics joins the list. This collection includes nineteen classroom-tested projects, eleven additional projects based on historical sources, three expository articles considering discrete mathematics topics in more depth, and two articles focused on pedagogy especially related to discrete mathematics.

Why is discrete mathematics only...

(pp. xi-xiv)
4. I Classroom-tested Projects
• The Game of “Take Away”
(pp. 3-6)
Mark MacLean

In this project, students play the game “Take Away” and conceive a winning strategy for the game. They then must give a careful written explanation of why their strategies work. I typically use this project in my discrete math class as an introduction to writing proofs. Students often struggle initially with the clarity of their mathematical exposition and with aiming their proofs at an appropriate audience. After completing this activity, we have a class discussion where we critique the students’ written explanations.

In most discrete math classes, students’ first proofs involve properties of even and odd integers. Since these properties...

• Pile Splitting Problem: Introducing Strong Induction
(pp. 7-10)
Bill Marion

In many textbooks in discrete mathematics there are numerous examples for teaching the Weak Form of the Principle of Mathematical Induction, but relatively few elementary problems for applying the Strong Form. What follows is a nice example to draw on when introducing the strong form. It can be presented as a classic puzzle, it has a number of variants and it is inherently recursive.

By introducing the problem (Pile Splitting) as a puzzle, the instructor can engage the students in the process of finding a general solution. She can, then, raise the question as to how they can demonstrate that...

• Generalizing Pascal: The Euler Triangles
(pp. 11-18)
Sandy Norman and Betty Travis

This project investigates generalizing Pascal’s Triangle to generate the coefficients appearing in the expansion of powers of multinomials of the form\$1 + x + {x^2} + ... - {x^k}\$. Students will initially use acounting pathsmethod and then relate the procedure to a generalized triangle, known as an Euler Triangle.

This activity is appropriate for mature Algebra and Precalculus students and can be accomplished in two or three one-hour class periods, depending on the backgrounds of the students. It fits nicely after a unit on binomial expansion. A peg board, with some type of peg or stud, can be used to string paths that can then...

• Coloring and Counting Rectangles on the Board
(pp. 19-30)
Michael A. Jones and Mika Munakata

We describe the Rectangles on the Board project, an adaptation of an activity for elementary and middle school students that appears in . Students are challenged to determine the coloring of the instructor’s 10☓10 board, given the restrictions that (1) all 100 squares are colored in one of four colors and (2) the colors form four rectangular regions, one in each color. Our extensions of this project involve counting, symmetry, geometry, and logical reasoning. For example, given the color of some squares, students infer the color of other squares based on geometry. In turn, they use logic and their understanding...

• Fun and Games with Squares and Planes
(pp. 31-44)
Maureen T. Carroll and Steven T. Dougherty

This project is intended to introduce students to the concepts of mutually orthogonal Latin squares and their relationship to finite affine planes. These topics are introduced in the first section. After describing how tic-tac-toe is played on an affine plane, the second section explores player strategies. Playing the game will help students understand the combinatorial and geometric notions described, and build geometric intuition for these objects.

Students must play the game in order to understand the strategy arguments. You can have them play against each other in class or turn in their game sheets as an exercise. While the last...

• Exploring Recursion with the Josephus Problem: (Or how to play “One Potato, Two Potato” for keeps)
(pp. 45-54)
Douglas E. Ensley and James E. Hamblin

The Josephus problem is addressed in many discrete mathematics textbooks as an exercise in recursive modeling, with some books (e.g.,  and ) even using it within the first few pages as an introductory problem to intrigue students. Since most students are familiar with the use of simple rhymes (like Eeny-meeny-miney-moe) for decision-making on the playground, they are comfortable with the physical process involved in this problem. For students who may wish to pursue this topic independently,  and  provide nice surveys and bibliographies, and the website  provides web-based tools for exploring the problem directly. The activities presented...

• Using Trains to Model Recurrence Relations
(pp. 55-60)
Benjamin Sinwell

This project gives students a hands-on approach to experiment with recurrence relations. By building trains using cars of different lengths students are given a concrete model with which to represent and better understand the concept of recursion.

A brief introduction to building trains of different lengths using cars of different lengths is all that is necessary for students to complete this assignment. For example, a train of length four can be made from two cars of length one and one car of length two in three different ways.

Cuisenaire rods or colored paper (cut to match the different train sizes)...

• Codon Classes
(pp. 61-64)
Brian Hopkins

This project explores an application of equivalence relations to bioinformatics. RNA can be considered as a string over a four-letter alphabet. Cells use triples of these bases to regulate the production of protein by signaling the sequence of amino acids. But there are far fewer than 64 amino acids; is there a system behind the redundancy?

Students examine the equivalence classes arising from six difference equivalence relations on the set of codons, from simple to somewhat intricate. Then they research the actual genetic code; it is “closest” to one of the simpler relations, although it is far from an exact...

• How to change coins, M&M’s, or chicken nuggets: The linear Diophantine problem of Frobenius
(pp. 65-74)
Matthias Beck

Let’s imagine that we introduce a new coin system. Instead of using pennies, nickels, dimes, and quarters, let’s say we agree on using 4-cent, 7-cent, 9-cent, and 34-cent coins. The reader might point out the following flaw of this new system: certain amounts cannot be exchanged, for example, 1, 2, or 5 cents. On the other hand, this deficiency makes our new coin system more interesting than the old one, because we can ask the question: “which amounts can be changed?” In the next section, we will prove that there are only finitely many integer amounts thatcannotbe exchanged...

• Calculator Activities for a Discrete Mathematics Course
(pp. 75-82)
Jean M. Horn and Toni T. Robertson

Using the symbolic capabilities of the TI-89 calculator can enhance discrete mathematics. Our mathematics and computer science majors take discrete mathematics as a post calculus course. Typical students come to the first class with their TI-89 in hand, both wanting and expecting to be able to use it. In response to their expectations we have developed a series of calculator activities which can be used to supplement and enhance some of the topics of discrete mathematics. They are designed to improve the basic understanding of course content, and to provide guidance for students as they transition from the concrete to...

• Bulgarian Solitaire
(pp. 83-92)
Suzanne Dorée

A player begins with coins arranged in piles. At each turn she rearranges the coins according to the following rule: remove the top coin from each pile, possibly eliminating piles, and form a new collected pile of coins. She repeats the process until she revisits a previously encountered arrangement, having reached a terminal cycle. Where are fixed points, if any? Are there any 2-cycles? Which states are cyclic? How can we visualize the process?

This process was dubbed “Bulgarian Solitaire” in a 1983Scientific Americancolumn by Martin Gardner , though I first saw it in Doug West and John...

• Can you make the geodesic dome?
(pp. 93-96)
Andrew Felt and Linda Lesniak

This project uses the geodesic dome made famous by Buckminster Fuller to extend students’ understanding of Eulerian graphs. Students are directed to build a dome and, after determining that the graphical representation of the dome contains neither an Euler cycle nor trail, to find the minimal number of repeated edges that are necessary to visit each edge.

I have done this construction six times with classes of up to 30 students, and each time the entire activity took less than two 50-minute class periods. There are many jobs to be done: corners to be held, rope to be strung through...

• Exploring Polyhedra and Discovering Euler’s Formula
(pp. 97-116)
Leah Wrenn Berman and Gordon Williams

The activities and exercises collected here provide an introduction to Euler’s formula, an introduction to interesting related topics, and sources for further exploration. While Euler’s formula applies to any planar graph, a natural and accessible context for the study of Euler’s formula is the study of polyhedra.

The article includes an introduction to Euler’s formula, four student activities, and two appendices containing useful information for the instructor, such as an inductive proof of Euler’s theorem and several other interesting results that may be proved using Euler’s theorem. Each activity includes a discussion of connections to discrete mathematics and notes to...

• Further Explorations with the Towers of Hanoi
(pp. 117-124)

This project is a supplement to the usual introduction to the Towers of Hanoi problems that appear in most discrete mathematics texts. Many concepts from discrete mathematics are discussed, including Hamiltonian paths, representation of integers in other bases, graph theory, and number theory.

Puzzles are a great way to pass time. However, they can also be an excellent source for mathematical applications. There are few puzzles more famous than the Towers of Hanoi. The point of this project is not to teach the students how to solve the puzzle, nor is it designed to teach a formula for the minimum...

• The Two Color Theorem
(pp. 125-130)
David Hunter

The purpose of this project is to help students prove that, under certain conditions, a map can be two-colored. There are at least four ways to approach this problem; the proofs vary in difficulty and use several techniques from discrete mathematics, including induction, structural induction, graphs, decision trees, and parity checks.

There are two parts to this project, and each part guides the students through two different proofs of the result. Part I is fairly straightforward and could be assigned as a homework problem or as an in-class group activity. Part II is more challenging and is suitable for a...

• Counting Perfect Matchings and Benzenoids
(pp. 131-142)
Fred J. Rispoli

The connection between perfect matchings and benzene was discovered by the German chemist Kekulé in the mid 1800s. Subsequently, chemists have learned that the number of perfect matchings contained in a molecular model is an important parameter related to chemical stability. Hence, counting perfect matchings has been an important problem in chemistry for over 50 years. However, counting perfect matchings in general graphs is a computationally difficult problem. Consequently, chemists and graph theorists have developed efficient counting methods for certain classes of graphs that arise in modeling special hydrocarbons called benzenoids. Many of these methods involve counting principles usually discussed...

• Exploring Data Compression via Binary Trees
(pp. 143-150)
Mark Daniel Ward

We investigate the Lempel-Ziv ’77 data compression algorithm by considering an analogous algorithm for efficiently embedding strings in binary trees. This project includes a discussion of this comparison with two optional addenda on error correction and decompression, followed by exercises and solutions.

Students in discretemathematics often have a dual interest in computer science. This project succinctly combines these two areas. Data compression can be viewed as a discrete mathematics topic with many ramifications for computer scientists. Students who have completed one or two semesters of computer science (in particular, who are familiar with trees) may be eager to implement the...

• A Problem in Typography
(pp. 151-158)
Larry E. Thomas

This project shows how a problem in computerized typesetting can be viewed as a problem in finding the shortest path from the beginning to the end of a weighted acyclic graph. Students are shown how to construct such a graph from the diagnostic information provided by Donald Knuth’s typesetting program, TEX.

This project uses a very special case of the general Knuth-Plass algorithm in order to make it manageable. The actual algorithm as implemented in  is general enough to handle unjustified text and mathematical expressions, but including all of the details would turn this into a course rather than...

• Graph Complexity
(pp. 159-162)
Michael Orrison

This project asks students to define, motivate, and explore an objective measure of the complexity of a graph.

This is an open-ended, capstone-like project designed to come at the end of the graph theory portion of a discrete mathematics course. I like this project because it provides a natural opportunity to discuss the general nature of mathematical research. Based on student feedback, I believe it is particularly successful because it gives students a genuine (and often unexpected) opportunity to express themselves in a mathematics course. Moreover, their insights have led, in some cases, to intriguing, worthwhile, and enjoyable independent student...

5. II Historical Projects in Discrete Mathematics and Computer Science
• Introduction
(pp. 165-168)
Janet Barnett, Guram Bezhanishvili, Hing Leung, Jerry Lodder, David Pengelley and Desh Ranjan

A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development in ancient cultures. By contrast, a course in finite mathematics is sometimes presented as a fast-paced news reel of facts and formulae, often memorized by the students, with the text offering only passing mention of the motivating...

• Binary Arithmetic: From Leibniz to von Neumann
(pp. 169-178)
Jerry M. Lodder

Gottfried Wilhelm Leibniz (1646–1716) is often described as the last universalist, having contributed to virtually all fields of scholarly interest of his time, including law, history, theology, politics, engineering, geology, physics, and perhaps most importantly, philosophy, mathematics and logic [1, 9, 11]. The young Leibniz began to teach himself Latin at the age of 8, and Greek a few years later, in order to read classics not written in his native language, German. Later in life, he wrote:

Before I reached the school-class in which logic was taught, I was deep into the historians and poets, for I began...

• Arithmetic Backwards from Shannon to the Chinese Abacus
(pp. 179-184)
Jerry M. Lodder

Recall that in the 1945 white paper “First Draft of a Report on the EDVAC” (Electronic Discrete Variable Automatic Computer), John von Neumann (1903–1957) advocated the use of binary arithmetic for the digital computers of his day. Vacuum tubes afforded thesemachines a speed of computation unmatched by other calculational devices, with von Neumann writing: “Vacuum tube aggregates . . . have been found reliable at reaction times as short as a microsecond . . . ” [7, p. 188].

Predating this, in 1938 Claude Shannon (1916–2001) published a ground-breaking paper “A Symbolic Analysis of Relay and Switching Circuits”...

• Pascal’s Treatise on the Arithmetical Triangle: Mathematical Induction, Combinations, the Binomial Theorem and Fermat’s Theorem
(pp. 185-196)
David Pengelley

Blaise Pascal (1623–1662) was born in Clermont-Ferrand in central France. Even as a teenager his father introduced him to meetings for mathematical discussion in Paris run by Marin Mersenne, who served as a primary conduit for transmitting mathematical ideas widely at that time, before the existence of any research journals. He quickly became involved in the development of projective geometry, the first in a sequence of highly creative mathematical and scientific episodes in his life, punctuated by periods of religious fervor. Around age twenty-one he spent several years developing a mechanical addition and subtraction machine, in part to help...

• Early Writings on Graph Theory: Euler Circuits and The Königsberg Bridge Problem
(pp. 197-208)
Janet Heine Barnett

In a 1670 letter to Christian Huygens (1629–1695), the celebrated philosopher and mathematician Gottfried W. Leibniz (1646–1716) wrote as follows:

I am not content with algebra, in that it yields neither the shortest proofs nor the most beautiful constructions of geometry. Consequently, in view of this, I consider that we need yet another kind of analysis, geometric or linear, which deals directly with position, as algebra deals with magnitude. [1, p. 30]

Known today as the field of ‘topology,’ Leibniz’s study of position was slow to develop as a mathematical field. As C. F. Gauss (1777–1855) noted...

• Counting Triangulations of a Convex Polygon
(pp. 209-216)
Desh Ranjan

In a 1751 letter to Christian Goldbach (1690–1764), Leonhard Euler (1707–1783) discusses the problem of counting the number of triangulations of a convex polygon. Euler, one of the most prolific mathematicians of all times, and Goldbach, who was a Professor of Mathematics and historian at St. Petersburg and later served as a tutor for Tsar Peter II, carried out extensive correspondence, mostly on mathematical matters. In his letter, Euler provides a “guessed” method for computing the number of triangulations of a polygon that hasnsides but does not provide a proof of his method. The method, if...

• Early Writings on Graph Theory: Hamiltonian Circuits and The Icosian Game
(pp. 217-224)
Janet Heine Barnett

Problems that are today considered to be part of modern graph theory originally appeared in a variety of different connections and contexts. Some of these original questions appear little more than games or puzzles. In the instance of the ‘Icosian Game’, this observation seems quite literally true. Yet for the game’s inventor, the Icosian Game encapsulated deep mathematical ideas which we will explore in this project.

SirWilliam Rowan Hamilton (1805–1865) was a child prodigy with a gift for both languages and mathematics. His academic talents were fostered by his uncle James Hamilton, an Anglican clergyman with whom he lived...

• Are All Infinities Created Equal?
(pp. 225-230)
Guram Bezhanishvili

Georg Ferdinand Ludwig Philip Cantor (1845–1918), the founder of set theory, and considered by many as one of the most original minds in the history of mathematics, was born in St. Petersburg, Russia in 1845. His parents, who were of Jewish descent, moved the family to Frankfurt, Germany in 1856. Georg entered theWiesbaden Gymnasium at the age of 15, and two years later began his university career at Zürich. In 1863 he moved to the University of Berlin, which during Cantor’s time was considered the world’s center of mathematical research. Four years later Cantor received his doctorate from the...

• Early Writings on Graph Theory: Topological Connections
(pp. 231-240)
Janet Heine Barnett

The earliest origins of graph theory can be found in puzzles and game, including Euler’s Königsberg Bridge Problem and Hamilton’s Icosian Game. A second important branch of mathematics that grew out of these same humble beginnings was the study of position (“analysis situs”), known today astopology¹. In this project, we examine some important connections between algebra, topology and graph theory that were recognized during the years from 1845–1930.

The origin of these connections lie in work done by physicist Gustav Robert Kirchhoff (1824–1887) on the flow of electricity in a network of wires. Kirchhoff showed how the...

• A Study of Logic and Programming via Turing Machines
(pp. 241-252)
Jerry M. Lodder

During the International Congress of Mathematicians in Paris in 1900 David Hilbert (1862–1943), one of the leading mathematicians of the last century, proposed a list of problems for following generations to ponder [8, p. 290–329], . On the list was whether the axioms of arithmetic are consistent, a question which would have profound consequences for the foundations of mathematics. Continuing in this direction, in 1928 Hilbert proposed the decision problem (das Entscheidungsproblem) [10, 11, 12], which asked whether there was a standard procedure that can be applied to decide whether a given mathematical statement is true. Both Alonzo...

• Church’s Thesis
(pp. 253-266)
Guram Bezhanishvili

In this project we will learn about both primitive recursive and general recursive functions. We will also learn about Turing computable functions, and will discuss why the class of general recursive functions coincides with the class of Turing computable functions. We will introduce the effectively calculable functions, and the ideas behind Alonzo Church’s (1903–1995) proposal to identify the class of effectively calculable functions with the class of general recursive functions, known as “Church’s thesis.” We will analyze Kurt Gödel’s (1906–1978) initial rejection of Church’s thesis, together with the work of Alan Turing (1912–1954) that finally convinced Gödel...

• Two-Way Deterministic Finite Automata
(pp. 267-274)
Hing Leung

In 1943, McCulloch and Pitts  published a pioneering work on a model for studying the behavior of the nervous systems. Following up on the ideas of McCulloch and Pitts, Kleene  wrote the first paper on finite automata, which proved a theorem that we now call Kleene’s theorem. A finite automaton can be considered as the simplest machine model in that the machine has a finite memory; that is, the memory size is independent of the input length. In a 1959 paper , Michael Rabin and Dana Scott presented a comprehensive study on the theory of finite automata, for...

6. III Articles Extending Discrete Mathematics Content
• A Rabbi, Three Sums, and Three Problems
(pp. 277-286)
Shai Simonson

We present a slice of discrete math history and connect it to three neat problems and their solutions. The solutions emphasize the value of exploring and tabulating data in order to help discover theorems. Often, the exploration not only helps discover theorems, but suggests a direction and idea for a proof.

The methodology is a model of how to use mathematical history and data patterns in teaching discrete mathematics. Proofs in discrete math tend to be constructive. Discovering patterns can suggest an algorithm, which in turn can suggest a proof. Furthermore, many theorems in discrete math make use of basic...

• Storing Graphs in Computer Memory
(pp. 287-292)
Larry E. Thomas

Many of the structures we use in discrete mathematics are nonlinear when they are drawn in the usual way on paper. For example, a graph will often have many edges leaving or entering a given vertex, with the number of edges changing from vertex to vertex. Drawing such a graph on paper could produce a rather messy diagram which takes two dimensions to represent. However, most graph algorithms used in problems of any size are executed on a computer which has only a linear (one-dimensional) memory. The question naturally arises: How do we represent an inherently two-dimensional object, such as...

• Inclusion-Exclusion and the Topology of Partially Ordered Sets
(pp. 293-302)
Eric Gottlieb

It is surprising and satisfying that the principle of inclusion-exclusion, a well-known combinatorial tool, is a special case of a more general phenomenon that also gives rise to the classical Möbius function of number theory. Specifically, the principle of inclusion-exclusion is equivalent to Möbius inversion on the Boolean lattice, while the classical Möbius function arises from Möbius inversion on the lattice of divisors. Generalized Möbius inversion was developed by Gian-Carlo Rota  in 1964.

To use Möbius inversion, one must be able to compute the value of what is known as thegeneralized Möbius function. This function provides a link...

7. IV Articles on Discrete Mathematics Pedagogy
• Guided Group Discovery in a Discrete Mathematics Course for Mathematics Majors
(pp. 305-312)
Mary E. Flahive

In this article we discuss the use of guided group discovery in Oregon State University’s discrete mathematics course for math majors. Since Fall 2003 this course has been taught at Oregon State using an ongoing modification of Kenneth P. Bogart’s successful group discovery method and notes , “Teaching Introductory Combinatorics by Guided Group Discovery.” Section 2 summarizes Ken’s notes and method, and Sections 3 and 4 respectively contain the adaptation of his notes and the implementation of his method at Oregon State.

Ken’s prototype was a small elective course in which the average entering student was very motivated to learn...

• The Use of Logic in Teaching Proof
(pp. 313-322)
Susanna S. Epp

Even rather simple proofs and disproofs are built atop a normally unexpressed substructure of great logical and linguistic complexity. For example, in  I described several of the many reasoning processes needed to establish the truth or falsity of the following statements: (1) The square of any rational number is rational; (2) For all real numbersaandb, if\$a>b\$then\${a^2}>{b^2}\$; and (3) For all real numbersx, ifxis irrational thenxis irrational. The article cites evidence that a significant number of students taking college mathematics courses do not bring with them an intuitive feeling...